<h2>Problem 257</h2>
<div style="color:#666;font-size:80%;">26 September 2009</div><br />
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<p>Given is an integer sided triangle ABC with sides a <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> b <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> c. 
(AB = c, BC = a and AC = b).<BR />
The angular bisectors of the triangle intersect the sides at points E, F and G (see picture below).
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<img src="project/images/p_257_bisector.gif" /><br />
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The segments EF, EG and FG partition the triangle ABC into four smaller triangles: AEG, BFE, CGF and EFG.<BR />
It can be proven that for each of these four triangles the ratio area(ABC)/area(subtriangle) is rational.<BR />
However, there exist triangles for which some or all of these ratios are integral.
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<p>
How many triangles ABC with perimeter<img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' />100,000,000 exist so that the ratio area(ABC)/area(AEG) is integral?
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